Production Function

Production Function

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Tanu Kathuria

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Production Function

Tanu Kathuria

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Production Function  

States the relationship between inputs and outputs Inputs – the factors of production classified as:  Land – all natural resources of the earth – not just ‘terra firma’!  Price paid to acquire land = Rent  Labour – all physical and mental human effort involved in production  Price paid to labour = Wages  Capital – buildings, machinery and equipment not used for its own sake but for the contribution it makes to production  Price paid for capital = Interest Tanu Kathuria

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Production Function Inputs Land Labour Capital

Process

Output

Product or service generated – value added

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Production Function Mathematical representation of the relationship:  Q = f (K, L, La)  Output (Q) is dependent upon the amount of capital (K), Land (L) and Labour (La) used 

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Analysis of Production Function: Short Run 

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In the short run at least one factor fixed in supply but all other factors capable of being changed Reflects ways in which firms respond to changes in output (demand) Can increase or decrease output using more or less of some factors but some likely to be easier to change than others Increase in total capacity only possible in the long run

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Analysis of Production Function: Short Run In times of rising sales (demand) firms can increase labour and capital but only up to a certain level – they will be limited by the amount of space. In this example, land is the fixed factor which cannot be altered in the short run.

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Analysis of Production Function: Short Run If demand slows down, the firm can reduce its variable factors – in this example it reduces its labour and capital but again, land is the factor which stays fixed.

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Analysis of Production Function: Short Run If demand slows down, the firm can reduce its variable factors – in this example, it reduces its labour and capital but again, land is the factor which stays fixed.

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Short-Run Changes in Production Factor Productivity Units of K Employed 8 7 6 5 4 3 2 1

37 42 37 31 24 17 8 4 1

60 64 52 47 39 29 18 8 2

Output Quantity (Q) 83 96 107 117 127 128 78 90 101 110 119 120 64 73 82 90 97 104 58 67 75 82 89 95 52 60 67 73 79 85 41 52 58 64 69 73 29 39 47 52 56 52 14 20 27 24 21 17 3 4 5 6 7 8 Units of L Employed

How much does the quantity of Q change, when the quantity of L is Tanu Kathuria increased?

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The Marginal Product of Labour

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The marginal product of labour is the increase in output obtained by adding 1 unit of labor but holding constant the inputs of all other factors Marginal Product of L: MPL= ∆Q/∆L (holding K constant) = δQ/δL Average Product of L: APL= Q/L (holding K constant) Tanu Kathuria

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Relationship Between Total, Average, and Marginal Product: Short-Run Analysis 

Total Product (TP) = total quantity of output

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Average Product (AP) = total product per total input

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Marginal Product (MP) = change in quantity when one additional unit of input used Tanu Kathuria

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Short-Run Analysis of Total, Average, and Marginal Product 

If MP > AP then AP is rising

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If MP < AP then AP is falling

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MP = AP when AP is maximized

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TP maximized when MP = 0

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Three Stages of Production in Short Run AP,MP

Stage I

Stage II

Stage III

APX Fixed input grossly underutilized; specialization and teamwork cause AP to increase when additional X is used

Specialization and teamwork continue to result in greater output when additional X is used; fixed input being properly utilized Tanu Kathuria

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MPX Fixed input capacity is reached; additional X causes output to fall 15

Law of Diminishing Returns

(Diminishing Marginal Product) Holding all factors constant except one, the law of diminishing returns says that: 

As additional units of a variable input are combined with a fixed input, at some point the additional output (i.e., marginal product) starts to diminish 

e.g. trying to increase labor input without also increasing capital will bring diminishing returns 

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Nothing says when diminishing returns will start to take effect, only that it will happen at some point All inputs added to the production process are exactly the same in individual productivity Tanu Kathuria

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Analysing the Production Function: Long Run 

The long run is defined as the period of time taken to vary all factors of production  By doing this, the firm is able to increase its total capacity – not just short term capacity  Associated with a change in the scale of production  The period of time varies according to the firm and the industry  In electricity supply, the time taken to build new capacity could be many years; for a market stall holder, the ‘long run’ could be as little as a few weeks or months!

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Analysis of Production Function: Long Run

In the long run, the firm can change all its factors of production thus increasing its total capacity. In this example it has doubled its capacity.

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Long-Run Changes in Production Returns to Scale Units of K Employed 8 7 6 5 4 3 2 1

37 42 37 31 24 17 8 4 1

60 64 52 47 39 29 18 8 2

83 78 64 58 52 41 29 14 3

Output Quantity (Q) 96 107 117 127 90 101 110 119 73 82 90 97 67 75 82 89 60 67 73 79 52 58 64 69 39 47 52 56 20 27 24 21 4 5 6 7 Units of L Employed

128 120 104 95 85 73 52 17 8

How much does the quantity of Q change, when the quantity of both L and K is increased? Tanu Kathuria

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Optimal Combination of Inputs 

Now we are ready to answer the question stated earlier, namely, how to determine the optimal combination of inputs

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As was said this optimal combination depends on the relative prices of inputs and on the degree to which they can be substituted for one another

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This relationship can be stated as follows:

MPL/MPK = PL/PK (or MPL/PL= MPK/PK) Tanu Kathuria

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An Isoquant Graph of Isoquant Y

7 6 5 4 3 2 1 0 1

2

3

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5

6

7

X

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Law of Diminishing Marginal Rate of Technical Substitution: Table 7.8 Input Combinations for Isoquant Q = 52 Combination L K A 6 2 B 4 3 C 3 4 D 2 6 E 2 8

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∆L -2 -1 -1 0

∆K 1 1 2 2

MRTS 2 1 1/2

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Law of Diminishing Marginal Rate of Technical Substitution continued Y 7

A

6 5

B

∆Y =- 2

4

C

∆X = 1 ∆Y = -1

3

D

∆X = 1

2

∆Y = -1

E

∆X = 2

1 0 2

3

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6

8

X

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Isocost Curves

Assume PL =$100 and PK =$200 Input Combinations Rs.1000 Budge Combination L K A 0 5 B 2 4 C 4 3 D 6 2 E 8 1 G 10 0

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Isocost Curve and Optimal Combination of L and K K 100L + 200K = 1000 5

“Q52” 10

L

Isocost andTanu isoquant curve for inputs L and 25 Kathuria

Expansion Path: the locus of points which presents the optimal input combinations for different isocost curves The long-run situation: both factors variable Units of capital (K)

Expansion path

300 TC = £60 000 TC = £20 000

O

TC = £40 000

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Units of labor (L)

Returns to Scale 

Let us now consider the effect of proportional increase in all inputs on the level of output produced

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To explain how much the output will increase we will use the concept of returns to scale

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Returns to Scale continued 

If all inputs into the production process are doubled, three things can happen: 

output can more than double 

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output can exactly double 

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increasing returns to scale (IRTS) constant returns to scale (CRTS)

output can less than double 

decreasing returns to scale (DRTS)

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Graphically, the returns to scale concept can be illustrated using the following graphs

Q

IRTS

Q

DRTS

CRTS

X,Y

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Q

X,Y

X,Y

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